📚 A-Level Mathematics: Methods and Applications of Non-Random Sampling | A-Level 数学：非随机抽样的方法与应用
In statistics, sampling is the process of selecting a subset of individuals from a population to estimate characteristics of the whole population. Non-random sampling, also called non-probability sampling, does not give every member of the population a known or equal chance of being selected. This article explores the main methods of non-random sampling and their practical applications, which are essential for A-Level Mathematics and beyond.
在统计学中，抽样是从总体中选取一部分个体来估计总体特征的过程。非随机抽样，也称为非概率抽样，并不给总体中的每个成员一个已知或相等的机会被选中。本文探讨非随机抽样的主要方法及其实际应用，这对 A-Level 数学及更高级的学习至关重要。
1. What is Non-Random Sampling? | 什么是非随机抽样？
Non-random sampling is a sampling technique where the samples are gathered in a process that does not give all individuals in the population equal chances of being selected. In contrast to random sampling, the selection depends on the subjective judgment of the researcher, convenience, or other non-random criteria.
非随机抽样是一种抽样技术，其收集样本的过程并不给予总体中所有个体相等的被选机会。与随机抽样相反，选择依赖于研究者的主观判断、便利性或其他非随机标准。
The key feature is that the probability of selection is unknown. This makes it difficult to use probability theory to quantify sampling error. However, non-random sampling is often faster, cheaper, and easier to implement, especially when a complete list of the population is not available.
其关键特征是选择概率是未知的。这使得难以用概率理论来量化抽样误差。然而，非随机抽样通常更快、更便宜且更容易实施，尤其是在没有完整总体名单的情况下。
2. Comparison with Random Sampling | 与随机抽样的比较
Random sampling methods, such as simple random sampling, systematic sampling, and stratified sampling, rely on chance and give every member of the population a known probability of being selected. This allows researchers to make statistical inferences about the population with measurable confidence.
随机抽样方法，如简单随机抽样、系统抽样和分层抽样，依赖于随机性，并给予总体中每个成员一个已知的被选概率。这使研究人员能够以可量化的置信度对总体进行统计推断。
Non-random sampling, on the other hand, is subjective and not based on probability. It is often used in exploratory research, pilot studies, and situations where randomness is impractical. The table below summarises key differences.
另一方面，非随机抽样是主观的，不基于概率。它常用于探索性研究、试点研究以及随机性不切实际的情况。下表总结了主要区别。
| Aspect | Random Sampling | Non-Random Sampling |
| Probability of selection | Known and equal (or adjustable) | Unknown, not equal |
| Sampling error | Can be quantified | Cannot be reliably quantified |
| Bias | Minimised | Likely to be present |
| Cost and time | Generally higher | Generally lower |
| Use of probability theory | Yes | No |
For A-Level Mathematics, it is important to know that statistical tests often assume random samples. When using non-random samples, conclusions must be drawn with caution.
对于 A-Level 数学，重要的是要知道统计检验通常假设随机样本。当使用非随机样本时，必须谨慎得出结论。
3. Convenience Sampling | 便利抽样
Convenience sampling involves selecting participants who are easiest to reach or access. For example, a researcher standing outside a shopping mall might ask passers-by to answer a questionnaire. This method is quick and inexpensive.
便利抽样涉及选择最容易接触或到达的参与者。例如，研究人员站在购物中心外，可能会请路过的行人回答问卷。这种方法快速且廉价。
However, convenience samples are often heavily biased because the sample is unlikely to represent the whole population. For instance, people who shop at a particular mall may have specific income levels or lifestyles.
然而，便利样本通常存在严重偏差，因为样本不太可能代表整个总体。例如，在某个商场购物的人可能具有特定的收入水平或生活方式。
- Advantages: Fast, low cost, easy to conduct.
- Disadvantages: High bias, poor representativeness.
- 优点：快速、低成本、易于实施。
- 缺点：偏差大、代表性差。
4. Quota Sampling | 配额抽样
Quota sampling is a non-random method that aims to ensure that certain characteristics of the population are represented in the sample. The researcher first identifies relevant strata, such as age, gender, or income, and then sets a quota for each group. For example, if the population is 50% female and 50% male, the sample must include 50 of each.
配额抽样是一种非随机方法，旨在确保总体的某些特征在样本中得到体现。研究者首先确定相关分层，如年龄、性别或收入，然后为每个群体设定配额。例如，如果总体是 50% 女性和 50% 男性，那么样本必须各包含 50 人。
Unlike stratified random sampling, quota sampling does not use random selection within each quota. The researcher may choose any convenient individuals to fill the quotas. This makes it cheaper, but it still risks bias from the choice of individuals.
与分层随机抽样不同，配额抽样在每个配额内不使用随机选择。研究者可以选择任何方便的个体来填补配额。这使其更便宜，但仍存在来自个体选择的偏差风险。
Quota sampling is popular in market research and opinion polls because it can produce fairly diverse samples without the cost of full randomisation.
配额抽样在市场研究和民意调查中很受欢迎，因为它可以在不完全随机化的成本下产生相当多样化的样本。
5. Judgement (Purposive) Sampling | 判断（目的）抽样
Judgement sampling, also called purposive sampling, relies on the expertise and judgement of the researcher to select individuals who are most representative of the population or most relevant to the study. For example, a researcher studying expert opinions on climate change might deliberately select climate scientists.
判断抽样，也称为目的抽样，依赖于研究者的专业知识和判断力来选择最能代表总体或与研究最相关的个体。例如，研究气候变化专家意见的研究者可能会特意选择气候科学家。
This method is useful when a small sample is needed and when the researcher knows exactly what characteristics are important. However, the validity of the results depends heavily on the researcher’s judgement, which can introduce bias.
当需要小样本并且研究者确切知道哪些特征重要时，这种方法非常有用。然而，结果的有效性在很大程度上取决于研究者的判断，这可能会引入偏差。
In A-Level contexts, judgement sampling is often mentioned when discussing qualitative research and case studies.
在 A-Level 考试中，判断抽样在讨论定性研究和案例研究时经常被提及。
6. Snowball Sampling | 雪球抽样
Snowball sampling is used when the population is difficult to access or hidden, such as individuals with a rare disease, drug users, or undocument workers. The researcher initially finds a small number of participants, who then recruit further participants from their acquaintances. This process continues, and the sample grows like a rolling snowball.
雪球抽样用于难以接触或隐藏的总体，例如罕见疾病患者、吸毒者或无证工人。研究者最初找到少量参与者，然后这些参与者从他们的熟人那里招募更多的参与者。这一过程持续进行，样本像滚雪球一样增长。
Snowball sampling is valuable for studying social networks and hidden populations. However, it suffers from a serious selection bias because participants are not independent; they are linked through social connections, and the sample may over-represent certain social groups.
雪球抽样对于研究社交网络和隐藏人群非常有价值。然而，它存在严重的选择偏差，因为参与者不是独立的；他们通过社会联系联系在一起，样本可能过度代表某些社会群体。
7. Volunteer (Self-Selected) Sampling | 志愿者（自愿）抽样
Volunteer sampling occurs when individuals choose to participate in a study by responding to an open invitation. Examples include online surveys, phone-in polls, or contestants in a reality show. The sample consists of people who are willing to participate, often because they have strong feelings about the topic.
志愿者抽样发生在个体通过对公开邀请作出回应而选择参与研究时。例子包括在线调查、电话投票或真人秀中的参赛者。样本由愿意参与的人组成，通常是因为他们对主题有强烈的感受。
This method is very easy to implement, but it leads to significant self-selection bias. The views of those who volunteer may be very different from the general population. For instance, a radio poll about a controversial issue may attract only those who are passionate, producing skewed results.
这种方法非常容易实施，但会导致显著的自选择偏差。自愿参与者的观点可能与一般总体非常不同。例如，关于一个有争议问题的电台投票可能只吸引那些充满热情的人，产生偏差的结果。
8. Advantages of Non-Random Sampling | 非随机抽样的优点
Non-random sampling methods offer several practical advantages in real-world research:
非随机抽样方法在实际研究中有几个实际优势：
- Low cost: They are generally cheaper than random sampling because they require fewer resources and less complex planning.
- Time efficiency: Data can be collected quickly, which is critical when decisions must be made promptly.
- Convenience: Researchers can easily access available subjects, especially in field settings.
- Feasibility: In some studies, random sampling is impossible because no complete list of the population exists.
- 低成本：通常比随机抽样便宜，因为所需的资源和复杂规划较少。
- 时间效率高：可以快速收集数据，这在需要迅速决策时至关重要。
- 便利性：研究人员可以轻松接触到可用的受试者，尤其是在实地环境中。
- 可行性：在某些研究中，随机抽样是不可能的，因为不存在完整的总体名单。
Because of these advantages, non-random sampling is widely used in exploratory research, pilot studies, and surveys where the aim is to gain initial insight rather than to make precise statistical predictions.
由于这些优点，非随机抽样广泛用于探索性研究、试点研究和以初步了解为目的而非精确统计预测的调查中。
9. Disadvantages of Non-Random Sampling | 非随机抽样的缺点
Despite its advantages, non-random sampling has serious limitations that must be understood.
尽管有优点，非随机抽样也具有必须被理解的严重局限性。
- Bias: The sample is likely not representative, leading to biased estimates of population parameters.
- No sampling error formula: Because selection probabilities are unknown, standard formulas for sampling error and confidence intervals cannot be applied.
- Limited generalisability: Results cannot be confidently generalised to the wider population.
- Risk of systematic errors: Subjective choices by the researcher can create systematic distortions.
- 偏差：样本很可能不具有代表性，导致对总体参数的估计有偏。
- 无抽样误差公式：由于选择概率未知，标准的抽样误差和置信区间公式不能使用。
- 推广性有限：结果不能自信地推广到更广泛的总体。
- 系统误差风险：研究者的主观选择可能造成系统性扭曲。
In A-Level Mathematics, you are expected to recognise these weaknesses and to suggest improvements, such as using random sampling where possible or acknowledging the limitations in conclusions.
在 A-Level 数学中，你需要识别这些弱点并提出改进建议，例如在可能的情况下使用随机抽样，或在结论中承认局限性。
10. Applications in Real-World Contexts | 在实际环境中的应用
Non-random sampling is widely used in many fields, including market research, public health, and social sciences. For example, convenience sampling is often used in university student surveys, while quota sampling is common in political polling. Judgement sampling is used in quality control when experts inspect selected products, and snowball sampling is used in epidemiological studies of hard-to-reach populations.
非随机抽样广泛应用于许多领域，包括市场研究、公共卫生和社会科学。例如，便利抽样常用于大学生调查，而配额抽样在政治民意调查中常见。判断抽样用于质量控制，专家检查选定的产品，雪球抽样用于难以接触人群的流行病学研究。
In the business world, companies may use volunteer sampling for customer feedback forms, though they know the respondents may not be representative of all customers. In education, teachers may use judgement sampling to select a few students for in-depth interviews.
在商业世界中，公司可能会使用志愿者抽样来收集客户反馈表，尽管他们知道受访者可能不能代表所有客户。在教育领域，教师可能会使用判断抽样来选择少数学生进行深入访谈。
Understanding when and why non-random sampling is used helps students appreciate the trade-off between cost and accuracy in real-world statistics.
理解非随机抽样在何时以及为何被使用，有助于学生体会现实世界中成本与准确性之间的权衡。
11. How to Choose a Sampling Method | 如何选择抽样方法
Choosing between random and non-random sampling depends on several factors: research objectives, the nature of the population, available resources, and the required level of accuracy.
在随机和非随机抽样之间进行选择取决于几个因素：研究目标、总体的性质、可用资源以及所需的准确度水平。
If the goal is to make precise statistical inferences, random sampling is preferable. If the goal is exploration, hypothesis generation, or when practical constraints are severe, non-random sampling may be acceptable. It is also possible to use a mixed approach, such as starting with a non-random sample for a pilot and then designing a random sample for the main study.
如果目标是进行精确的统计推断，随机抽样更可取。如果目标是探索、假设生成，或当实际约束严重时，非随机抽样可能是可接受的。也可以使用混合方法，例如先从非随机样本开始进行试点，然后为主要研究设计随机样本。
When using non-random sampling, it is crucial to report the method clearly and to discuss the potential biases that may affect the findings.
使用非随机抽样时，必须清楚报告方法并讨论可能影响研究结果的潜在偏差。
12. Summary | 总结
Non-random sampling methods include convenience, quota, judgement, snowball, and volunteer sampling. They offer practical advantages such as low cost and speed, but they introduce bias and limit the ability to quantify sampling error.
非随机抽样方法包括便利抽样、配额抽样、判断抽样、雪球抽样和志愿者抽样。它们提供低成本和高速度等实际优势，但会引入偏差并限制量化抽样误差的能力。
For A-Level Mathematics, it is essential to understand these methods, their strengths and weaknesses, and when they are appropriate. You should be able to compare them with random sampling and to evaluate the reliability of conclusions drawn from non-random samples. This understanding will not only help you in exams but also in real-world data analysis.
对于 A-Level 数学，理解这些方法、它们的优缺点以及何时适用至关重要。你应该能够将它们与随机抽样进行比较，并评估从非随机样本得出结论的可靠性。这种理解不仅会在考试中帮助你，也会在现实世界的数据分析中帮助你。
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